AMC 10 · 2013 · #21
Grade 7 countingTwo non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is N. What is the smallest possible value of N?
Pick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Two sequences of nonnegative integers each go up (or stay level) and follow the Fibonacci-style rule: from the third term on, each term is the sum of the two before it. Their first terms are not the same, yet both sequences have the same seventh term, called N. Find the smallest N this is possible for.
Givens: Each sequence is non-decreasing and made of nonnegative integers; From the third term on, each term = sum of the previous two terms; The two sequences have different first terms; Both sequences share the same seventh term N
Unknowns: The smallest possible value of the shared seventh term N
Understand
Restated: Two sequences of nonnegative integers each go up (or stay level) and follow the Fibonacci-style rule: from the third term on, each term is the sum of the two before it. Their first terms are not the same, yet both sequences have the same seventh term, called N. Find the smallest N this is possible for.
Givens: Each sequence is non-decreasing and made of nonnegative integers; From the third term on, each term = sum of the previous two terms; The two sequences have different first terms; Both sequences share the same seventh term N
Plan
Primary tool: #14 Extreme Principle
Secondary: #4 Introduce a Variable, #5 Look for a Pattern, #2 Make a Systematic List
The phrase 'smallest possible value' is a direct call for the Extreme Principle: push N as low as it can go while everything still holds. First name the two starting terms a and b so the seventh term becomes a clean formula in a and b. That formula turns 'two sequences with the same seventh term' into 'two different (a, b) pairs giving the same number', and the Extreme Principle then chases the smallest such number allowed by the ordering rule a <= b.
Execute — Answer: C
5.OA.B.3 Step 1 Write the seventh term with variables
- Call the first two terms a and b.
- The rule 'each term is the sum of the two before' builds the rest: a, b, a+b, a+2b, 2a+3b, 3a+5b, and finally the seventh term 5a+8b.
- So every allowed sequence has seventh term N = 5a + 8b, where a and b are its first two terms.
💡 Naming the two starting numbers lets the fixed adding rule carry them all the way to a single formula for the seventh term.
6.EE.B.6 Step 2 Two sequences, one value of N
- Let one sequence start with a, b and the other with a', b'.
- Both hit the same seventh term means 5a + 8b = 5a' + 8b'.
- Rearranging, 5(a' - a) = 8(b - b').
- Since the first terms differ (a != a'), the two sides are not zero.
💡 Setting the two seventh-term formulas equal turns the whole puzzle into one relationship between the two starting pairs.
6.NS.B.4 Step 3 The smallest step between solutions
- In 5(a' - a) = 8(b - b'), the left side is a multiple of 5 and the right a multiple of 8.
- Because 5 and 8 share no common factor, a' - a must be a multiple of 8 and b - b' a multiple of 5.
- The smallest nonzero change is a' - a = 8 and b - b' = 5.
- So from any valid pair (a, b) the next one is (a + 8, b - 5): raise the first term by 8, drop the second by 5.
💡 Because 5 and 8 have no common factor, the first term can only jump in whole steps of 8 while the second slides by 5.
7.EE.B.4 Step 4 Apply the ordering rule and minimize
- Both sequences must be non-decreasing, so a <= b and a' <= b'.
- The second pair needs a + 8 <= b - 5, which means b >= a + 13.
- Now minimize N = 5a + 8b with a >= 0 and b >= a + 13.
- Both coefficients are positive, so make a and b as small as allowed: take a = 0, which forces b >= 13, so b = 13.
💡 Pushing both starting numbers to their smallest legal values drives the seventh term as low as it can go.
4.NBT.B.4 Step 5 Check both sequences and read off N
- With a = 0, b = 13 the first sequence is 0, 13, 13, 26, 39, 65, 104.
- The partner pair is (0+8, 13-5) = (8, 8), giving 8, 8, 16, 24, 40, 64, 104.
- Both are non-decreasing, start with different first terms (0 versus 8), and both reach 104 as the seventh term.
- So the smallest possible N is 104, which is (C).
💡 One explicit pair of sequences ending in 104 proves the bound is really reachable, not just hoped for.
5.OA.B.3 Call the first two terms a and b. The rule 'each term is the sum of the two befo 6.EE.B.6 Let one sequence start with a, b and the other with a', b'. Both hit the same se 6.NS.B.4 In 5(a' - a) = 8(b - b'), the left side is a multiple of 5 and the right a multi 7.EE.B.4 Both sequences must be non-decreasing, so a <= b and a' <= b'. The second pair n 4.NBT.B.4 With a = 0, b = 13 the first sequence is 0, 13, 13, 26, 39, 65, 104. The partner Review
Reasonableness: The bound is tight from both sides: N = 5a + 8b with the forced gap b >= a + 13 cannot dip below 5(0) + 8(13) = 104, and the concrete pair of sequences 0,13,...,104 and 8,8,...,104 shows 104 is achieved with different first terms. The smaller options 55 and 89 are Fibonacci numbers reachable by only one natural pair, so they cannot come from two different starts; 104 is the first value expressible as 5a + 8b in two ordered ways.
Alternative: Skip the divisibility step and just list values of 5a + 8b for 0 <= a <= b in increasing order, watching for the first value that appears twice with different a. The pairs (0,13) and (8,8) both give 104 before any smaller repeat shows up, confirming N = 104.
CCSS standards used (min grade 7)
5.OA.B.3Generate two numerical patterns using two given rules and identify relationships (Building each sequence from the add-the-previous-two rule to reach the seventh term 5a + 8b.)6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the first terms a, b and setting the two seventh-term expressions equal.)6.NS.B.4Find greatest common factor and least common multiple of two numbers (Using gcd(5,8)=1 to show the first term must change in steps of 8 and the second in steps of 5.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Turning the non-decreasing rule into b >= a + 13 and minimizing N = 5a + 8b under it.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Computing the terms of both sequences and confirming each seventh term equals 104.)
⭐ Turn the seventh term into the formula 5a + 8b, then push the two starting numbers as low as the rising-order rule allows to find the smallest shared value.
⭐ Turn the seventh term into the formula 5a + 8b, then push the two starting numbers as low as the rising-order rule allows to find the smallest shared value.
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